If a storm window has an area of 315 square inches, what are the dimensions if the window is 6 inches higher than it is wide (w x h)?
Area = length x width
Length = width + 6
A = lw
l = w + 6
315 = (w+6)(w)
315 = w^2 +6w
0 = w^2 + 6w - 315
0 = (w+21)(w-15)
w+21 = 0 w - 15 = 0
w = -21 w = 15
discard the -21 because we don't measure distance in negative numbers.
w = 15
l = 21
Multiply to be sure they equal the area.
w(w+6) = 315
since 315 = 15*21, ...
Let's assume the width of the storm window is "w" inches.
According to the problem, the height of the storm window is 6 inches more than its width. So, the height can be represented as "w + 6" inches.
The area of a rectangle is calculated by multiplying its length (width) by its height. Given that the area of the storm window is 315 square inches, we can set up the following equation:
w * (w + 6) = 315
Now, let's solve this equation step by step:
1. Expand the equation:
w^2 + 6w = 315
2. Rearrange the equation in standard quadratic form:
w^2 + 6w - 315 = 0
3. Factorize the quadratic equation:
(w - 15)(w + 21) = 0
Setting each factor equal to zero:
w - 15 = 0 or w + 21 = 0
Solving for width:
w = 15 or w = -21
Since width cannot be negative in this context, the width of the storm window is 15 inches.
To find the height, substitute the width value into the equation:
height = w + 6 = 15 + 6 = 21 inches
Therefore, the dimensions of the storm window are 15 inches in width (w) and 21 inches in height (h).
To find the dimensions of the storm window, we can set up an equation based on the given information.
Let's assume the width of the window is 'w' inches. According to the problem, the height of the window is 6 inches more than its width. So, the height can be expressed as 'w + 6' inches.
To find the area of the window, we use the formula for the area of a rectangle: Area = width × height.
Given that the area of the window is 315 square inches, we can write the equation as:
315 = w × (w + 6)
Now, we need to solve this quadratic equation to determine the value of 'w' and ultimately find the dimensions of the window.
Let's proceed by expanding the equation:
315 = w^2 + 6w
Rearranging the equation to bring all the terms to one side:
w^2 + 6w - 315 = 0
Now, we can either factor this quadratic equation or use the quadratic formula to solve for 'w'.
Let's use the quadratic formula:
The quadratic formula states that for an equation of the form ax^2 + bx + c = 0, the solutions for x can be found using the formula:
x = (-b ± sqrt(b^2 - 4ac)) / (2a)
In our equation, a = 1, b = 6, and c = -315. Substituting these values into the quadratic formula, we get:
w = (-6 ± sqrt(6^2 - 4(1)(-315))) / (2(1))
Calculating further:
w = (-6 ± sqrt(36 + 1260)) / 2
w = (-6 ± sqrt(1296)) / 2
w = (-6 ± 36) / 2
Therefore, we have two possible solutions for 'w':
w = (36 - 6) / 2 = 30 / 2 = 15
w = (-36 - 6) / 2 = -42 / 2 = -21
Since the width of a window cannot be negative, we disregard the negative solution. Therefore, the width of the storm window is 15 inches.
To find the height, we can substitute this value back into the equation 'w + 6':
Height = 15 + 6 = 21 inches
So, the dimensions of the window are 15 inches (width) by 21 inches (height).